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I am reading about L-functions of Elliptic curves from Milne's notes. The definition of the L-function of an Elliptic curve defined over $\mathbb Q$ is $$L(E,s)=\prod_{p\text{ good}} (1-a_pp^{-s}+p^{1-2s})^{-1}\prod_{p\text{ bad}}(1-a_pp^{-s})^{-1}$$ where for good primes $p$, we have that $a_p=\begin{cases}\#\tilde{E}(\mathbb F_p)&\text{good reduction at }p\\0 &\text{ additive reduction at }p\\1& \text{multiplicative split reduction at } p\\-1&\text{multiplicative non-split reduction at }p\end{cases}$

I am unable to understand the rationale behind the choice of the coefficients of the $p$-factors of the $L$-functions. The only possible explanation that I have found out till now is that the functional equation is satisfied because of the choice.

I would prefer a clearer and more natural explanation for the coefficients. Some historical background is also appreciated.

Grobber
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1 Answers1

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This is the so-called Hasse-Weil L-function of an elliptic curve over $\mathbb{Q}$, which is defined as the product over its local $L$-series: $$ L(E,s):=\prod_p L_p(E,s)^ {-1}, $$ related to the Hasse-Weil Zeta function, i.e., by $$ Z_{E,\mathbb{Q}}(s)=\frac{\zeta(s)\zeta(s-1)}{L(E,s)} $$ Regarding your question: there is really a lot of mathematics going on behind this definition. This has been explained on this site and at Mathoverflow already quite good. A starting point could be this question:

$L$-functions of elliptic curves over $\mathbb{Q}$

Dietrich Burde
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  • Let $E : y^2=x^3-ax-b$ and $R = \mathbb{Z}[x,y]/(y^2-x^3-ax-b)$. Then $Z_{E(\mathbb{Q})}(s) = \zeta_R(s) = \prod_{\mathfrak{m}} \frac{1}{1-(N\mathfrak{m})^{-s}}$ where $N\mathfrak{m}=#\ R/\mathfrak{m}$, $\mathfrak{m} = (p,f(x),g(x,y))$ where $P \in E(\mathbf{F}p)$, $f \in \mathbf{F}_p[x]$ the minimal polynomial of $x_P$ and $g \in \mathbf{F}_p[x]/(f(x))[y]$ the minimal polynomial of $y_P$. If we have unique factorization of ideals $\mathfrak{a}$ with $R/\mathfrak{a}$ finite, then $\prod{\mathfrak{m}} \frac{1}{1-(N\mathfrak{m})^{-s}} = \sum_\mathfrak{a} (N\mathfrak{a})^{-s}$ – reuns Nov 13 '17 at 15:31
  • I didn't say wikipedia isn't correct, I'm just saying this is the clearly the motivation of Hasse-Weil zeta functions. – reuns Nov 13 '17 at 15:34
  • I see. So this is also not mentioned in the linked question, that's what you want to say. Yes, there are more motivations, indeed. I did not want to summarize it myself. The link is just a help to read further. – Dietrich Burde Nov 13 '17 at 15:34
  • A step to obtain what I wrote is : For a fixed $p$ let $M_k = # {\mathfrak{m}, N\mathfrak{m} = p^k}$ then $\log \prod_{\mathfrak{m}} \frac{1}{1-(N\mathfrak{m})^{-s }} = -\sum_{k=1}^\infty M_k \log(1-p^{-sk}) = \sum_{k=1}^\infty \sum_{l=1}^\infty M_k\frac{p^{-slk}}{l}$ $ = \sum_{m=1}^\infty p^{-sm} \sum_{k | m}\frac{M_k}{m/k} =\sum_{m=1}^\infty p^{-sm} \frac{# E(\mathbf{F}{p^m})}{m}$ ie. $ # E(\mathbf{F}{p^m}) = \sum_{k | m} k M_k$ which is explained for example in Milne's book : points on $E(\mathbf{F}{p^k})$ are morphisms $\mathbb{Z}[E] \to \mathbf{F}_p[E] \to \mathbf{F}{p^k}$. – reuns Nov 13 '17 at 15:57